12 UNBOUNDED NORMAL OPERATORS 423 We have Sp(Nn) a Sp(JV) for all n, for if N - £7 is a bijective mapping of dom(JV) onto E, then each of the operators Nn - £/EM is a bijective (and hence bicontinuous, by the closed graph theorem) mapping of En onto EM. From this point onwards, the proof of the first assertion merely repeats that of (15.11.5), having regard to the facts that Sp(JV) is closed (15.12.11) and that (Nn - a/EjT1 is a continuous normal operator on En for all a £ Sp(7Vn), and the proposition (15.12.10). The assertion about the point-spectra is immediate, and the eigenspaces of N are determined as in (15.11.6). To show that the residual spectrum of N is empty, it is enough to show that the same is true for each Nn, so that we reduce to the case where N is continuous; then, by using the decomposition of (15.11.3), we may assume that N is multiplication by the class of the function lc in L£(K, /*), where K is a compact subset of C, and /A is a positive measure on K. We have then to show that, if the measure ju is not concentrated at the point 0, the set of functions CH+£#(£), where g E JSfc(K, ju), is dense in JS?c(K, JM). For this it is sufficient, by (6.3.1), to show that if a function h e JS?£(K, ju) is such that J &(0*(0 4*(0 = 0 for all g e JSfc(K, ju)> then h is ju-negligible. But this is clear, by taking #(Q = C/z(0> in view of the hypothesis on ju. Finally, it is clear that if N is self-adjoint, then so is each Nn, hence Sp(Nn) c R for all n, and therefore Sp(N) c= R. Conversely, if Sp(N) is contained in R, then Sp(7Vn) c R for all rc, hence Nn is self-adjoint for all n by (15.11.7), and therefore so is N by (15.12.8). (15.12.13) Propositions (15.12.8) and (15.12.12) essentially serve to reduce problems relating to unbounded normal operators to the same problems for bounded (i.e., continuous) normal operators. In particular, let/:C-»C be universally measurable. We shall show that for each unbounded normal operator N we can define an unbounded normal operator /(N), having properties analogous to those of (15.11.1). Suppose first that the function/ is bounded; then, with the notation of (15.12.8),/(NJ is a continuous normal operator on En (15.11.1) and moreover, \\f(Nn)\\ g sup |/(Q| for all n £eC (15.11.8). The normal operator on E whose restriction to each En is/(Nn) is therefore continuous (15.10.8.1), and it is this operator which is denoted by f(N), It is immediately seen that if g is another function in ^C(C)» then (/+ 0)(N) =/(#) + 8(N)> (M(N) =/(N)#(N) and/(JV) = (/(N))*, by virtue of (15.11.1.1). Now let/: C ~* C be any universally measurable function. For each n ;> 0, let An be the universally measurable subset of C consisting of the complex numbers £ such that n ^ |/(Q| < n -f 1, so that (AB) is a partition of C. It follows from above that Pn = (p&n(N) is an orthogonal projection in E (15.5.3.1), and E is the Hilbert sum of the closed subspaces HM = Pn(E); also,"1.